Abstract
In this paper a method to construct Kripke models for subtheories of constructive set theory is introduced that uses constructions from classical model theory such as constructible sets and generic extensions. Under the main construction all axioms except the collection axioms can be shown to hold in the constructed Kripke model. It is shown that by carefully choosing the classical models various instances of the collection axioms, such as exponentiation, can be forced to hold as well. The paper does not contain any deep results. It consists of first observations on the subject, and is meant to introduce some notions that could serve as a foundation for further research.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aczel P.: The type-theoretic interpretation of constructive set theory. In: Troelstra, A.S., van Dalen, D. (eds) Logic Colloquium 1977, pp. 55–66. North-Holland, Amsterdam (1978)
Aczel, P. : The type-theoretic interpretation of constructive set theory: choice principles. In: Macintyre, A. (ed.) The L.E.J. Brouwer Centenary Symposium, pp. 1–40. Amsterdam (1982)
Aczel P. : The type-theoretic interpretation of constructive set theory: inductive definitions. In: Marcus, R.B. (eds) Logic, Methodology and Philosophy of Science VII, pp. 17–49. North-Holland, Amsterdam (1986)
Aczel, P., Rathjen, M.: Notes on constructive set theory, Manuscript (2007)
Gambino N.: Heyting-valued interpretations for constructive set theory. Ann. Pure Appl. Logic 137(1–3), 164–188 (2006)
Kunen K.: Set Theory—An Introduction to Independence Proofs, Studies in Logic and the Foundations of Mathematics. Elsevier, Amsterdam (1980)
Lubarsky R.: Independence results around constructive ZF. Ann. Pure Appl. Logic 132(2–3), 209–225 (2005)
Lubarsky R.: CZF and second order arithmetic. Ann. Pure Appl. Logic 141(1–2), 29–34 (2006)
Myhill J.: Constructive set theory. J. Symbolic Logic 40, 347–382 (1975)
van Oosten J.: Realizability: An Introduction to its Categorical Side. Elsevier, Amsterdam (2008)
Rathjen M.: The disjunction and related properties for constructive Zermelo-Fraenkel set theory. J. Symbolic Logic 70(4), 1233–1254 (2005)
Rathjen M., Tupailo S.: Characterizing the interpretation of set theory in Martin-Löf type theory. Ann. Pure Appl. Logic 141(3), 442–471 (2006)
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Iemhoff, R. Kripke models for subtheories of CZF . Arch. Math. Logic 49, 147–167 (2010). https://doi.org/10.1007/s00153-009-0164-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-009-0164-0